Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-3y &= 5 \\ 3x+9y &= -9\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $9y = -3x-9$ Divide both sides by $9$ to isolate $y$ $y = {-\dfrac{1}{3}x - 1}$ Substitute this expression for $y$ in the first equation. $-5x-3({-\dfrac{1}{3}x - 1}) = 5$ $-5x + x + 3 = 5$ Simplify by combining terms, then solve for $x$ $-4x + 3 = 5$ $-4x = 2$ $x = -\dfrac{1}{2}$ Substitute $-\dfrac{1}{2}$ for $x$ back into the top equation. $-5( -\dfrac{1}{2})-3y = 5$ $\dfrac{5}{2}-3y = 5$ $-3y = \dfrac{5}{2}$ $y = -\dfrac{5}{6}$ The solution is $\enspace x = -\dfrac{1}{2}, \enspace y = -\dfrac{5}{6}$.